Question

Directions: Answer true or false. If false, provide a counterexample.

Answer 1

Natural numbers are closed under division: false.

A set is closed under a certain operation if the results of that operation are always inside that set.

So, if natural numbers were closed under division, the division of two natural numbers would always be a natural number.

You have plenty of counterexamples, to pick one you may divide any odd number by 2: 5/2 is not a natural number.

Negative numbers are closed under addition: true.

Let $m,n$ be two positive numbers. So, $-m,-n$ are two negative numbers. Their sum is

$(-m)+(-n)=-(m+n)$

And since $m+n$ is positive, we deduce that $-(m+n)$ is negative, so the sum of two negative numbers is still negative.

Prime numbers are closed under subtraction: false.

This would mean that the subtraction of two primes is also a prime. Again, there are many counterexamples: 7 is prime and so is 3, but their difference 7-3 is 4, which is not prime.